Red Square

A problem from the 1999 Russian Mathematical Olympiad:

Each cell of a 50×50 square is colored in one of four colors. Show that there exists a square which has cells of the same color as it directly above, directly below, directly to the left, and directly to the right of it (though not necessarily adjacent to it).

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A 50×50 square contains 2500 cells. At least a quarter of these, 625 cells, must be of the same color, say red. Of these 625 red squares, at most 50 can be the uppermost red squares in their columns; at most 50 can be the lowermost red squares in their columns; at most 50 can be the leftmost red squares in their rows; and at most 50 can be the rightmost red squares in their rows. Together these make up at most 200 squares. The remaining 425 (or more) red squares have none of these distinctions, so each must have a red square directly above, directly below, directly to the left, and directly to the right of it. From Titu Andreescu and Zuming Feng, Mathematical Olympiads 1999-2000: Problems and Solutions from Around the World, 2002.

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September 20, 2020September 19, 2020 | Puzzles

A Perfect Square

An old problem from the Soviet Mathematical Olympiad:

Find the 4-digit number aabb that is a perfect square.

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The smallest 4-digit perfect square is 1024 (32²). The largest is 9801 (99²). And aabb = 1000a + 100a + 10b + b = 1100a + 11b = 11(100a + b). So 11 is one of the factors of aabb, and aabb is of the form (11k)². That leaves only six candidates to check (33², 44², 55², 66², 77², 88²), and it turns out the answer is 88² = 7744. (Thanks, Danesh.)

September 15, 2020September 14, 2020 | Puzzles

Competing Squares

competing squares

A square has been inscribed in each of two congruent isosceles right triangles. Which square is larger?

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Divide each figure into a grid of triangles. This shows that the square on the left occupies 2/4 of its triangle, while the square on the right occupies 4/9 of its triangle. So the square on the left is about 12.5 percent larger. (Duane Detemple and Sonia Harold, “A Round-Up of Square Problems,” Mathematics Magazine 69:1 [February 1996], 15-27.)

September 10, 2020October 11, 2020 | Puzzles

Black and White

tchepizhni chess problem

This problem, by V. Tchepizhni, won fifth prize in the Bohemian Centenary Tourney of 1962. It’s four puzzles in one:

(a) The diagrammed position.
(b) Turn the board 90 degrees clockwise.
(c) Turn the board 180 degrees.
(d) Turn the board 270 degrees clockwise.

Each of the resulting positions presents a helpmate in two: Black moves first, then White, then Black, then White, the two sides cooperating to reach a position in which Black is checkmated. What are the solutions?

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(a) 1. c1=R Rxg5 2. Rc3 Bc2# (b) 1. b5 Bc3+ 2. Kc5 Ba5# (c) 1. b3 Rb4 2. f6 Bf7# (d) 1. g2 Bf4+ 2. Kf2 Bh2# (From John M. Rice, An A.B.C. of Chess Problems, 1970.)

September 5, 2020September 4, 2020 | Puzzles

A Stressful Game

A puzzle by Ezra Brown and James Tanton:

Three gnomes sit in a circle. An evil villain puts a hat on each gnome’s head. Each hat is either rouge or puce, the color chosen by the toss of a coin. Each gnome can see the color of his companions’ hats but not of his own.

At the villain’s signal, all three gnomes must speak at once, each either guessing the color of his own hat or saying “Pass.” If at least one of them guesses correctly and none guesses incorrectly, all three gnomes will live. But if any of them guesses incorrectly, or if all three pass, they’ll all die.

They may not communicate in any way during the game, but they can confer beforehand. How can they arrange a 75 percent chance that they’ll survive?

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Each gnome will pass unless he sees that both of his companions are wearing hats of the same color. In that case he will guess the opposite color. This produces a win in six of the eight possible arrangements of hats. (Ezra Brown and James Tanton, “A Dozen Hat Problems,” Math Horizons 16:4 [April 2009], 22-25.)

August 27, 2020August 26, 2020 | Puzzles

Black and White

glass chess problem

By F.G. Glass, 1903. White to mate in four moves. (Easier than it sounds.)

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If the white king just circles the pawn, all Black’s moves are forced: 1. Kf1 Kd2 2. Kf2 Kd1 3. Ke3 Ke1 4. Rc1#

August 15, 2020August 14, 2020 | Puzzles

Inside Business

A puzzle from MIT Technology Review, September-October 1999: A popular pastime at Bell Labs in the 1970s was to identify a word that contains a given string of letters — for example, OOKKEE is found in BOOKKEEPER. Michael Foster found English words that contain HIPE, HCR, and UFA. What are they?

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ARCHIPELAGO, WITCHCRAFT, and MANUFACTURE.

August 6, 2020August 5, 2020 | Language · Puzzles

Podcast Episode 300: Lateral Thinking Puzzles

Here are six new lateral thinking puzzles — play along with us as we try to untangle some perplexing situations using yes-or-no questions.

See full show notes …

June 22, 2020June 29, 2020 | Podcast · Puzzles

Black and White

healey chess problem

By Frank Healey. White to mate in two moves.

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With the quiet 1. Qa1! White threatens to push his pawn, giving mate. Black can stop this only with 1. … c4, and then the queen can mate from a5. From A Collection of Two Hundred Chess Problems, 1866.

March 26, 2020March 26, 2020 | Puzzles

Secret Message

Jonathan Swift’s Journal to Stella, a collection of 65 letters written to his friend Esther Johnson, contains some puzzling passages, such as this one:

“He gave me al bsadnuk lboinlpl dfaonr ufainfbtoy dpionufnad, which I sent him again by Mr. Lewis.”

How should the obscured phrase be read?

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“A bank bill for fifty pound.” It’s a null cipher in which alternate letters are ignored. (From David Kahn’s The Codebreakers.)

March 24, 2020March 23, 2020 | Literature · Puzzles